Found problems: 85335
There are two-way flights between some of the $2017$ cities in a country, such that given two cities, it is possible to reach one from the other. No matter how the flights are appointed, one can define $k$ cities as "special city", so that there is a direct flight from each city to at least one "special city". Find the minimum value of $k$.
The circle $\Gamma$ and the line $\ell$ have no common points. Let $AB$ be the diameter of $\Gamma$ perpendicular to $\ell$, with $B$ closer to $\ell$ than $A$. An arbitrary point $C\not= A$, $B$ is chosen on $\Gamma$. The line $AC$ intersects $\ell$ at $D$. The line $DE$ is tangent to $\Gamma$ at $E$, with $B$ and $E$ on the same side of $AC$. Let $BE$ intersect $\ell$ at $F$, and let $AF$ intersect $\Gamma$ at $G\not= A$. Let $H$ be the reflection of $G$ in $AB$. Show that $F,C$, and $H$ are collinear.
What is the least real value of the expression $\sqrt{x^2-6x+13} + \sqrt{x^2-14x+58}$ where $x$ is a real number?
$
\textbf{(A)}\ \sqrt {39}
\qquad\textbf{(B)}\ 6
\qquad\textbf{(C)}\ \frac {43}6
\qquad\textbf{(D)}\ 2\sqrt 2 + \sqrt {13}
\qquad\textbf{(E)}\ \text{None of the above}
$
Find all real numbers $x,y$ satisfying the following system of equations
\begin{align*}
x^4 +y^4 & =17(x+y)^2 \\
xy & =2(x+y).
\end{align*}
We call a subset $B$ of natural numbers [i]loyal[/i] if there exists natural numbers $i\le j$ such that $B=\{i,i+1,\ldots,j\}$. Let $Q$ be the set of all [i]loyal[/i] sets. For every subset $A=\{a_1<a_2<\ldots<a_k\}$ of $\{1,2,\ldots,n\}$ we set
\[f(A)=\max_{1\le i \le k-1}{a_{i+1}-a_i}\qquad\text{and}\qquad g(A)=\max_{B\subseteq A, B\in Q} |B|.\] Furthermore, we define \[F(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} f(A)\qquad\text{and}\qquad G(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} g(A).\] Prove that there exists $m\in \mathbb N$ such that for each natural number $n>m$ we have $F(n)>G(n)$. (By $|A|$ we mean the number of elements of $A$, and if $|A|\le 1$, we define $f(A)$ to be zero).
[i]Proposed by Javad Abedi[/i]
Find all polynomials $P$ with real coefficients such that $\forall{x\in\mathbb{R}}$ we have:
\[ P(2P(x))=2P(P(x))+2(P(x))^2. \]
Quadratic function $f(x)=ax^2+bx+c$ satisfies that:
$(1)\forall x\in\mathbb{R},f(x-4)=f(2-x),f(x)\geq x$;
$(2)\forall x\in(0,2),f(x)\leq\left(\frac{x+1}{2}\right)^2$;
$(3)\min\limits_{x\in\mathbb{R}}f(x)=0$
Find the maximum of $m(m>1)$, satisfying:
There exists $t\in\mathbb{R}$, as long as $x\in[1,m]$, then $f(x+t)\leq x$.
For every integer $m\ge 1$, let $\mathbb{Z}/m\mathbb{Z}$ denote the set of integers modulo $m$. Let $p$ be a fixed prime and let $a\ge 2$ and $e\ge 1$ be fixed integers. Given a function $f\colon \mathbb{Z}/a\mathbb{Z}\to \mathbb{Z}/p^e\mathbb{Z}$ and an integer $k\ge 0$, the $k$[i]th finite difference[/i], denoted $\Delta^k f$, is the function from $\mathbb{Z}/a\mathbb{Z}$ to $\mathbb{Z}/p^e\mathbb{Z}$ defined recursively by
\begin{align*}
\Delta^0 f(n)&=f(n)\\
\Delta^k f(n)&=\Delta^{k-1}f(n+1)-\Delta^{k-1}f(n) & \text{for } k=1,2,\dots.
\end{align*}
Determine the number of functions $f$ such that there exists some $k\ge 1$ for which $\Delta^kf=f$.
[i]Holden Mui[/i]
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions
\[f(1+xy)-f(x+y)=f(x)f(y) \quad \text{for all } x,y \in \mathbb{R},\]
and $f(-1) \neq 0$.
Find the smallest natural number $n$ which has the following properties:
a) Its decimal representation has a 6 as the last digit.
b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $n$.
Let $x_1,x_2,\cdots,x_n$ be positive real numbers such that $x_1+x_2+\cdots+x_n=1$ $(n\ge 2)$. Prove that\[\sum_{i=1}^n\frac{x_i}{x_{i+1}-x^3_{i+1}}\ge \frac{n^3}{n^2-1}.\]here $x_{n+1}=x_1.$
A beam of length $ a $ is suspended horizontally with its ends on two parallel ropes equal $ b $. We turn the beam through an angle $ \varphi $ around a vertical axis passing through the center of the beam. By how much will the beam rise?
Regarding a natural number $n$, it is stated that the number $n^2$ has $7$ as the second to last digit. What is the last digit of $n^2$?
Let $ABC$ be an acute angled triangle with $H$ as its orthocentre. For any point $P$ on the circumcircle of triangle $ABC$, let $Q$ be the point of intersection of the line $BH$ with line $AP$. Show that there is a unique point $X$ on the circumcircle of triangle $ABC$ such that for every $P$ other than $B,C$, the circumcircle of $HPQ$ passes through $X$.
For $n ,p \in N^*$ , $ 1 \le p \le n$, we define
$$ R_n^p = \sum_{k=0}^p (p-k)^n(-1)^k C_{n+1}^k $$
Show that: $R_n^{n-p+1} =R_n^p$ .
Let $n \geq 2$ be an integer. Let $P(x_1, x_2, \ldots, x_n)$ be a nonconstant $n$-variable polynomial with real coefficients. Assume that whenever $r_1, r_2, \ldots , r_n$ are real numbers, at least two of which are equal, we have $P(r_1, r_2, \ldots , r_n) = 0$. Prove that $P(x_1, x_2, \ldots, x_n)$ cannot be written as the sum of fewer than $n!$ monomials. (A monomial is a polynomial of the form $cx^{d_1}_1 x^{d_2}_2\ldots x^{d_n}_n$, where $c$ is a nonzero real number and $d_1$, $d_2$, $\ldots$, $d_n$ are nonnegative integers.)
[i]Proposed by Ankan Bhattacharya[/i]
Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$.
[i]Proposed by Japan[/i]
How many ordered triples $(a,b,c)$ of integers with $1\le a\le b\le c\le 60$ satisfy $a\cdot b=c$?
In a chess tournament, each player played at most one game against each other, and the number of games played by each player is not less than the set natural number $ n $. Prove that it is possible to divide players into two groups $ A $ and $ B $ in such a way that the number of games played by each player of group $ A $ with players of group $ B $ is not less than $ n/2 $ and at the same time the number of games played by each player of the $ B $ group with players of the $ A $ group was not less than $ n/2 $.
Each coefficient $a_n$ of the power series \[a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots = f(x)\] has either the value of $1$ or the value $0.$ Prove the easier of the two assertions:
(i) If $f(0.5)$ is a rational number, $f(x)$ is a rational function.
(ii) If $f(0.5)$ is not a rational number, $f(x)$ is not a rational function.
In the acute-angled triangle $ABC$ with circumcircle $\omega$ and orthocenter $H$, points $D$ and $E$ are the feet of the perpendiculars from $A$ onto $BC$ and from $B$ onto $AC$ respecively. Let $P$ be a point on the minor arc $BC$ of $\omega$ . Points $M$ and $N$ are the feet of the perpendiculars from $P$ onto lines $BC$ and $AC$ respectively. Let $PH$ and $MN$ intersect at $R$. Prove that $\angle DMR=\angle MDR$.
Let $A,B,C$ be three distinct points in space, $(A)$ the sphere with center $A$ and radius $r$. Let $E$ be the set of numbers $R>0$ for which there is a sphere $(H)$ with center $H$ and radius $R$ such that $B$ and $C$ are outside the sphere, and the points of the sphere $(A)$ are strictly inside it.
(a) Suppose that $B$ and $C$ are on a line with $A$ and strictly outside $(A)$. Show that $E$ is nonempty and bounded, and determine its supremum in terms of the given data.
(b) Find a necessary and sufficient condition for $E$ to be nonempty and bounded
(c) Given $r$, compute the smallest possible supremum of $E$, if it exists.
Natural numbers $a$ and $b$ are relatively prime. Prove that the greatest common divisor of $(a+b)$ and $(a^2+b^2)$ is either $1$ or $2$.
Show that the sum $S=5+5^2+5^3+…+5^{2016}$ is divisible by $31$
What is the largest integer $n$ such that $n$ is divisible by every integer less than $\sqrt[3]{n}$?